In the circle with center $O$ and diameters $AC$ and $BD$, the angle $AOD$ measures $54$ degrees. What is the measure, in degrees, of angle $AOB$? [asy] draw(circle((0,0),1)); label("O",(0,0),SE); draw((-1,0)--(1,0)); draw((0.5877852522924731291,0.80901699437)--(-0.5877852522924731291,-0.80901699437)); label("B",(-1,0),W); label("D",(1,0),E); label("A",(0.5877852522924731291,0.80901699437),NE); label("C",(-0.5877852522924731291,-0.80901699437),SW); dot((0,0)); dot((-0.5877852522924731291,-0.80901699437)); dot((0.5877852522924731291,0.80901699437)); dot((-1,0)); dot((1,0)); [/asy]
Solution: Since $AC$ and $BD$ are line segments that intersect at point $O$, angle $AOD$ and angle $AOB$ are supplementary angles and their angle measures must add up to $180$ degrees. Since angle $AOD$ measures $54$ degrees, the measure of angle $AOB$ must be $180 - 54 = \boxed{126}$ degrees.